Bolo Toss – an example of center of mass motion.

Maybe this game has always been popular, but I see people playing while tailgating for football games. The basic idea is that you toss these two balls connected by strings (a bolo) at a ladder. Points are scored based on where the bolo lands. Here is an example:

Question: Does the center of mass of the two-ball system have a constant acceleration? Well, it should – but does it? Time for some video analysis with Tracker. (Oh, if you want to analyze this yourself, you can download the video from youtube).

First, you might need to adjust the coordinate system a little in each frame – I tried to keep the camera still, but I was just holding it with my hand). As a bonus, Tracker’s autotracking feature worked pretty well for this video. After get data for both balls, it looks like this (note: I could not figure out how to get Tracker to show the plots of two different objects at the same time – I am sure it can be done and I am just an idiot).

Here is a plot of the x-position of the balls (and of the center of mass of the balls – in green):

I also fit a linear function to the center of mass data and I get:

This is what you would expect. For the system, the only force is the gravitational force, which is in the y-direction. This means that there should be no change in the velocity in the x-direction (for the system).

And here is the similar data for the y-direction:

I added a quadratic fit to this data:

My scale is way off. However, this quadratic data does fit quite nicely. Don’t you think?

Modeling motion in python

Now let me go the other way. Let me model the motion of two balls attached by a string as it is thrown in the air. The motion of each ball should be similar to the data above. First, let me draw a force diagram for these balls as they are being thrown:

Key point: both balls have the same mass, so the same gravitational force. Both balls are attached by the same string, so the tension on the two balls will be the same magnitude, but opposite direction. How will I model this in python? I am going to cheat, well, not really. I am going to model the string as a super-stiff spring. There are two ways I could take into account the string. I could call it a force of constraint. In that case, the string would be whatever force it needs to be to keep the two masses the same distance apart. Or, if it is a spring, it will reach an equilibrium position and essentially do the same thing.

Here you can see that both balls have motions are rather squiggly. This is because of the spring. I could fix this – with two different methods. First, I could add a dampening force to the motion to make the spring behave more like a string. Or, I could start the two balls off with better initial conditions. If in these initial conditions, the balls are at an equilibrium distance from each other, there will be no oscillations with respect to the center of mass. One important thing to notice is that even though the two balls have squiggles, the center of mass motion is a parabola (constant acceleration).

Why does the center of mass behave?

It is all about systems. If I take both balls as the system, then there is only the external gravitational force on the system. The string exerts a force on one ball, but it exerts the same (but opposite) force on the other ball. The effect is that the system moves just like a point mass being thrown.

If you look at just one of the masses, then there are two external forces on that mass: the string and gravity. The combination of these two forces is what makes the ball move in a non-parabolic motion.

What if I just look at the motion of the balls with respect to the center of mass? Here is the real data from the video showing the trajectory with respect to center of mass.